Started some notes about the 2-spin partition function.overleaf

1 files changed, 14 insertions, 0 deletions

diff --git a/stokes.tex b/stokes.tex
index 91f7f18..70a7ed7 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -392,6 +392,20 @@ separatrix of a third. This means that when the imaginary energies of two
 critical points are brought to the same value, their surfaces of constant
 imaginary energy join.
 
+\begin{equation}
+  \begin{aligned}
+    Z(\beta)
+    &=\int_{S^{N-1}}dx\,e^{-\beta H(x)}
+    =\int_{\mathbb R^N}dx\,\delta(x^2-N)e^{-\beta H(x)} \\
+    &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12\beta x^TJx-\lambda(x^Tx-N)} \\
+    &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12x^T(\beta J+\lambda I)x+\lambda N} \\
+    &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\det(\beta J+\lambda I)}}e^{\lambda N} \\
+    &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\prod_i(\beta\lambda_i+\lambda)}}e^{\lambda N} \\
+    &=(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac12\sum_i\log(\beta\lambda_i+\lambda)} \\
+    &\simeq(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac N2\int d\lambda'\,\rho(\lambda')\log(\beta\lambda'+\lambda)} \\
+  \end{aligned}
+\end{equation}
+
 \subsection{Pure \textit{p}-spin}
 
 \begin{equation}